Buddhism and 'The Hard Problem'
The 'Hard Problem' is that there is no explanation of how neurological activity could turn into qualitative subjective experience. Not only is there no explanation, but no one has any idea of what such an explanation would look like.

Even worse (for the reductionists) there's a growing suspicion that The Hard Problem is in fact what's known as a `category error' or 'category mistake'. In other words, no explanation of subjective experience in terms of physico-chemical brain activity is possible. Another dimension of reality is required. There is a huge, vast, gaping explanatory gap between brain processes ('neural correlates') and mental experience.

Mental experience cannot be explained in terms of causes and conditions nor structures of the brain. There are correlations, but no explanations.

This shouldn't be surprising if we remember teachings on sunyata, that the three irreducible aspects of all phenomena are:

Mental imputation is associated with, but cannot be reduced to the two other aspects. The mind is an irreducible feature of reality.

If you were to put it in Buddhist terminology, I suppose you would say that the brain is not a valid basis of imputation for the mind (and any case what would be the entity that was doing the imputing?).

The short circuit from mind to particle physics
The flaw at the bottom level is that there seems to be a direct connection to the top level which doesn't go through the hierarchy. The links between the behaviour of fundamental particles and the activities of the mind are suspiciously close.

There are fascinating parallels between what physicists have learned about sub-atomic physics and the Buddhist ideas of emptiness and dependent origination. In addition, as Einstein famously remarked "The most incomprehensible thing about the universe is that it is comprehensible"

The problem of emergence
The flaw in the middle is the lack of equivalence in both directions (aka the problem of emergence). You can logically go down the hierarchy, but not up it. From observing the behaviour of sub-atomic particles you could not even deduce the existence of turnips, let alone brains, though close examination of both turnips and brains would lead to the discovery of their constituent particles.

`Nothing but' is untrue, because there is some mysterious `extra ingredient' being added as you go up the hierarchy.

So we will have to replace the phrase `is nothing but' by `is understandable in terms of', and do a few more corrections as well.

The mind is NOT understandable in terms of the brain. The brain is understandable in terms of biological systems.
Biological systems are understandable in terms of chemical interactions.
Chemical interactions are understandable in terms of physical interactions.
Physical interactions are understandable in terms of mathematics.
Mathematics is a product of the mind.

Now, forget the bottom and top and just look at the middle level of the hierarchy. Can you see the mysterious missing ingredient? Where does the understanding come from?

Let's turn the middle layers upside down and rephrase using Buddhist terminology...

Physical interactions are the basis of imputation of chemical reactions.
Chemical interactions are the basis of imputation of biological systems
...etc.

So even at the more mundane intermediate levels of this supposedly objective reductionist hierarchy, mental imputation is present all the time.

Mind is an irreducible aspect of reality inseparably intertwined with all levels of the physical universe.

Buddhists have nothing to fear from Science
Maybe Buddhists need to put more effort into combining arguments from science and Buddhism to demolish the deluded view of Reductionism ( aka Materialism aka Physicalism aka Naturalism).

Buddhism is grounded in philosophy, and has nothing to fear from science, not even from Darwin's Universal Acid which corrodes the 'faith-based' religions.

The bleak, deluded view of Materialist/Physicalist Reductionism is not only a major obstacle to the spiritual progress of those who (often reluctantly ) suffer from it, but it also generates fear, aggression and denial in those who oppose it but don't know how to argue against it. This denial and aggression against Materialism manifests as anti-science, irrationalism, bigotry, Creationism, Biblical literalism and is quite possibly a contributory factor to Jihadism.

But there's no reason for Buddhists to adopt the same head-in-the-sand turn-the-clock-back approach. Buddha didn't tell us to criticise and examine his teachings for no reason. He knew that his Dharma rested on unassailable metaphysical foundations.